Dictionary:Phase characteristics

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FIG. P-2. (a) Phase characterization of wavelets having the same amplitude spectrum. (b) Minimum-phase wavelet and its phase spectrum: (1–0.8z)2(1+0.5z)2=1–0.6z–0.71z2+0.24z3+0.16z4. (c) Linear phase: (1–0.8z)(0.8–z)((1+0.5z)(0.5+z)=0.4+0.18z–1.25z2+0.18z3+0.4z4. (d) Maximum phase: (0.8–z)2(0.5+z)2=0.16+0.24z–0.71z2–0.6z3+z4. (e) Zero phase: 0.4z–2+0.18z–1–1.25+0.18z+0.4z2. The zero-phase wavelet is anticipatory, that is, it begins before time zero. Phase curves depend on the time reference. Other mixed-phase wavelets can also be made from these component doublets. (f) Z-plane plot of the roots of the autocorrelation function for the foregoing, all of which have the same autocorrelation: xy(z)=(1–0.8z)2(0.8–z)2(1+0.5z)2(0.5+z)2. In a more general case roots may be complex.

1. Of the set of all those wavelets, filters, or systems that have the same amplitude spectrum or autocorrelation, particular members can be characterized by their phase spectra (phase as a function of frequency). (They can also be characterized in other ways, for example by the location of their roots in the z-domain; see Figure P-2.) The principal feature of minimum phase is that the energy arrives as early as possible. The phase of a minimum-phase wavelet is smaller and its energy builds up faster (i.e., it is minimum delay) than for any other causal wavelet with the same amplitude spectrum (or same autocorrelation). A two-term wavelet (or doublet) [a,b] is minimum phase (minimum delay) if |a|>|b|.

Any wavelet may be represented as the convolution of doublets and a wavelet is minimum phase if all of its doublet factors are minimum phase. For example, the z-transform of a wavelet might be (6+zz2), which can be expressed as (3–z)(2+z), each of which is minimum phase; hence the wavelet is minimum phase. Minimum phase is sometimes expressed as having all roots outside the unit circle in the z-plane, or as having no zeros in the right half of the Laplace transform S-plane. A maximum-phase or maximum-delay doublet [a,b] has |a|<|b|. Maximum-phase wavelets have all their roots inside the unit circle in the z-plane. For a linear-phase wavelet, the phase-frequency plot is linear. If its intercept is nπ (where n is any integer), such a wavelet is symmetrical.

A zero-phase wavelet has phase identically zero; it is symmetrical about zero but is not causal.